Theorem bnj1071 34989

Description: Technical lemma for bnj69 35022. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1071.7	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1071	⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Proof of Theorem bnj1071

Step	Hyp	Ref	Expression
1		bnj1071.7	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj923 34780	. 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
3		nnord 7804	. 2 ⊢ (𝑛 ∈ ω → Ord 𝑛)
4		ordfr 6321	. 2 ⊢ (Ord 𝑛 → E Fr 𝑛)
5	2, 3, 4	3syl 18	1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ∖ cdif 3894  ∅c0 4280  {csn 4573   E cep 5513   Fr wfr 5564  Ord word 6305  ωcom 7796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-ss 3914  df-uni 4857  df-tr 5197  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309  df-on 6310  df-om 7797

This theorem is referenced by:  bnj1030  34999  bnj1133  35001